This question was previously asked in

JKSSB JE CE 2015 Official Paper

Option 4 : Maxwell's Reciprocal deflection theorem

Building Material & Concrete Technology

15599

20 Questions
20 Marks
25 Mins

**Explanation**

**a) **__Maxwell-Betti Theorem:__

Maxwell-Betti's law of real work is a basic theorem in the structural analysis.

Using this theorem, it can be established that the flexibility coefficients in compatibility equations, formulated to solve indeterminate structures by the flexibility method, form a symmetric matrix and this will reduce the number of deflection computations.

The Maxwell-Betti law also has applications in the construction of influence line diagrams for statically indeterminate structures.

The Maxwell-Betti law applies to any stable elastic structure (a beam, truss, or frame, for example) on unyielding supports and at a constant temperature, states.

Let’s suppose two systems of forces

According to the Maxwell-Betti theorem,

∴ Pi × y1 = Pj × y2

Where, y = Deflection and P is the force

From the above, we can say that in any beam the deflection at any point 'i' due to load Pi, at any other point 'y' is the same as the deflection at' 'i' due to the same load Pj applied at 'j'

**b) **__Castigliano’s Second Theorem:__

In a structure, for a linear and elastic material, constant temperature, and unyielding support, the first partial derivative of total strain energy with respect to any force is equal to the deflection at that point in the direction of the force.

**c) **__Castigliano’s First Theorem:__

In a structure, for a linear and elastic material, constant temperature, and unyielding support, the first partial derivative of total strain energy with respect to any displacement component is equal to the external force applied at that point in the direction of the displacement.